Advanced Micro 1 Lecture 6: Producer Theory Nicolas Schutz

Nicolas Schutz

Producer Theory

1 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

2 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

3 / 18

Maintain the following assumptions: firms maximize profit firms are price-takers (and face linear prices) These assumptions are very strong (and a substantial body of literature relaxes them).

Nicolas Schutz

Producer Theory

4 / 18

There are k commodities, so the commodity space is Rk . A netput vector (or a production plan) is an element of Rk . Example: k = 3 and z = (−5, 0, 2). The firm transforms 5 unit of product 1 into 2 units of product 3. Let Z ⊆ Rk (Z , ∅) be the set of feasible netput vectors (also called the production-possibility set).

Nicolas Schutz

Producer Theory

5 / 18

Some possible assumptions on the shape of Z:

Definition 1 Free disposal: If z ∈ Z and z0 ≤ z, then z0 ∈ Z. Convexity: Z is convex. Strict convexity: Z is strictly convex, i.e., ˚ (∀z, z0 ∈ Z)(∀a ∈ (0, 1))(z , z0 ⇒ az + (1 − a)z0 ∈ Z) Closedness: Z is closed. Increasing returns to scale (IRS): If z ∈ Z and a ≥ 1, then az ∈ Z. Decreasing returns to scale (DRS): If z ∈ Z and 0 ≤ a ≤ 1, then az ∈ Z. Constant returns to scale (CRS): If z ∈ Z and a ≥ 0, then az ∈ Z.

Nicolas Schutz

Producer Theory

6 / 18

Define the firm’s profit-maximization problem at price vector p >> 0 (PMP(p)): max p.z s.t. z ∈ Z. z

Definition 2 The set of solutions of PMP(p) is called the set of optimal netput vectors, or the set of optimal production plans. We denote it by Z? (p). Besides, let  π(p) = sup p · z : z ∈ Z . π is called the profit function.

Nicolas Schutz

Producer Theory

7 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

8 / 18

Need to impose restrictions on the shape of Z to ensure that the PMP has a solution. Clearly, we’ll need closedness. Also:

Lemma 1 Suppose that Z has IRS. Then, for every p >> 0, if π(p) > 0, then π(p) = ∞.

Nicolas Schutz

Producer Theory

9 / 18

An ugly (but useful) property:

Definition 3 x ∈ Rk is an accumulation point of sequence (xn )n∈N ∈ (Rk )N if there exists a subsequence of (xn )n∈N which converges to x. N Z satisfies the recession-cone (RC) property if for every sequence   (zn )n∈N ∈ Z zn belongs to such that limn→∞ ||zn || = ∞, every accumulation point of ||zn || n∈N

R−k .

Proposition 1 Suppose Z is closed. Then, a solution to the PMP exists for every p >> 0 if and only if Z satisfies RC.

Nicolas Schutz

Producer Theory

10 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

11 / 18

Proposition 2 Suppose Z is closed and satisfies RC. Then: (i) For all λ > 0, Z∗ (λp) = Z∗ (p) and π(λp) = λπ(p). (ii) If Z is convex, then Z∗ (p) is convex for each p. If Z is strictly convex, then Z∗ (p) is a singleton for each p. (iii) π is convex. If Z∗ (p) is a singleton for all p, then denote by z∗ (p) the function associated with this singleton-valued correspondence.

Nicolas Schutz

Producer Theory

12 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

13 / 18

Proposition 3 If Z is closed, strictly convex and satisfies RC, then z∗ (p) and π(p) are continuous.

Nicolas Schutz

Producer Theory

14 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

15 / 18

Hotelling’s lemma:

Proposition 4 If Z is closed, strictly convex and satisfies RC, then π is differentiable and ∇π(p) = z∗ (p).

Nicolas Schutz

Producer Theory

16 / 18

Plan 1

Introduction

2

Existence of a solution

3

Properties of the PMP

4

Continuity

5

Differentiability of the profit function

6

The own-price effect for firms

Nicolas Schutz

Producer Theory

17 / 18

Proposition 5 Let z ∈ Z∗ (p) and z0 ∈ Z∗ (p0 ). Then (p − p0 ) · (z − z0 ) ≥ 0. The law of supply/demand for firms? Prices and quantities move in the same direction.

Nicolas Schutz

Producer Theory

18 / 18